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infinity' - in each instance. Now imagine the tractrix to be
revolved about the line XOX'. The double-trumpet surface is
generated; for reasons we need not go into (it has constant
negative curvature) it is called a pseudo-sphere. If on this
surface we draw the four-sided figure with two equal sides and
two right angles as before, using geodesies, we find that the
hypothesis of the acute angle is realized.
Thus the hypotheses of the right angle, the obtuse angle, and
the acute angle respectively are true on a Euclidean plane, a
sphere, and a pseudosphere respectively, and in all cases
'straight lines' are geodesies or extrema. Euclidean geometry is a
limiting, or degenerate, case of geometry on a sphere, being
attained when the radius of the sphere becomes infinite.
Instead of constructing a geometry to fit the Earth as human
beings now know it, Euclid apparently proceeded on the as-
sumption that the Earth is flat. If Euclid did not, his prede-
cessors did, and by the time the theory of 'space', or geometry,
reached him the bald assumptions which he embodied in his
postulates had already tak;en on the aspect of hoary and im-
mutable necessary truths, revealed to mankind by a higher
intelligence as the veritable essence of all material things. It
took over 2,000 years to knock the eternal truth out of geome-
try, and Lobatchewsky did it.
To use Einstein's phrase, Lobatchewsky challenged an axiom.
Anyone who challenges an 'accepted truth* that has seemed
necessary or reasonable to the great majority of sane men for
2,000 years or more takes bis scientific reputation, if not his
life, in his hands. Einstein himself challenged the axiom that
two events can happen in different places at the same time, and
by analysing this hoary assumption was led to the invention of
the special theory of relativity. Lobatchewsky challenged the
assumption that Euclid's parallel postulate or, what is equi-
valent, the hypothesis of the right angle, is necessary to a con-
sistent geometry, and he backed his challenge by producing a
system of geometry based on the hypothesis of the acute angle
in which there is not one parallel through a fixed point to a
given straight line but two. Neither of Lobatchewsky's parallels
meets the line to which both are parallel, nor does any straight